PHYSICAL REVIEW A 89, 063805 (2014)

Two-mode squeezed states in cavity optomechanics via engineering of a single reservoir

M. J. Woolley1 and A. A. Clerk2

1School of Engineering and Information Technology, UNSW Canberra, ACT, 2600, Australia2Department of Physics, McGill University, Montreal, QC, Canada H3A 2T8

(Received 9 April 2014; published 11 June 2014)

We study theoretically a three-mode optomechanical system where two mechanical oscillators are indepen-dently coupled to a single cavity mode. By optimized two-tone or four-tone driving of the cavity, one canprepare the mechanical oscillators in an entangled two-mode squeezed state, even if they start in a thermal state.The highly pure, symmetric steady state achieved allows the optimal fidelity of standard continuous-variableteleportation protocols to be achieved. In contrast to other reservoir engineering approaches to generatingmechanical entanglement, only a single reservoir is required to prepare the highly pure entangled steady state,greatly simplifying experimental implementation. The entanglement may be verified via a bound on the Duaninequality obtained from the cavity output spectrum. A similar technique may be used for the preparation of ahighly pure two-mode squeezed state of two cavity modes, coupled to a common mechanical oscillator.

DOI: 10.1103/PhysRevA.89.063805 PACS number(s): 42.50.Dv, 03.67.Bg, 42.50.Lc, 85.85.+j

I. INTRODUCTION

The generation and detection of entangled states of macro-scopic mechanical oscillators is an outstanding task in thestudy of mechanical systems in the quantum regime [1].There exist a number of proposals for the generation ofsuch states [2–6]. Perhaps most promising amongst these areapproaches based on reservoir engineering [7–10], wherebythe dissipation is engineered such that the steady state ofthe dissipative dynamics is the desired target state. Theseproposals are highly attractive from an experimental pointof view, requiring relatively minor modifications of existingexperimental configurations [11–14].

Here, we propose an approach for generating highly pure,highly entangled two-mode squeezed states of two mechanicaloscillators via coupling to a driven cavity mode. The two-modesqueezed state is the simultaneous vacuum of two nonlocalbosonic operators (so-called Bogoliubov modes) [15]. Hence,it is possible to prepare a mechanical two-mode squeezed stateby cooling these two modes. This can be achieved using twoindependent reservoirs (see, e.g. Refs. [7–9]). Here, however,we show that the same goal can be achieved using just a singlereservoir. By making the Bogoliubov modes nondegenerate,they will couple to different frequency components of a singlereservoir (that is, the damped cavity). Since the Bogoliubovtransformation preserves the difference in number operators,this simply corresponds to a frequency difference of the twomechanical oscillators. Viewed differently, one can say thatwe are exploiting the coherent dynamics of the mechanicaloscillators to effectively cool both Bogoliubov modes. Ulti-mately, our protocol simply involves appropriately weightedand detuned two-tone or four-tone driving of the coupled cavitymode.

The approach we take here may be regarded as the coherentfeedback [16] analog of our measurement-based approach tothe same task [5], and is related to a recent proposal for thepreparation of a quantum squeezed state of a single mechanicaloscillator [17]. As compared with the measurement-basedapproach for entanglement generation in Ref. [5], the purity ofthe steady state achieved here is greater and the implementationis greatly simplified. Further, in contrast to the proposal

of Ref. [8], the steady state is a (highly pure) two-modesqueezed state rather than an entangled mixed state. Therefore,using the steady state as an Einstein-Podolsky-Rosen (EPR)channel, the optimal teleportation fidelity for a given amountof entanglement can be achieved via the standard continuous-variable teleportation protocol, without the need for additionallocal operations [18,19]. In contrast to the proposal of Ref. [9](also described in the supplement to Ref. [8]), the two-modesqueezed state is generated here using only one, rather thantwo, auxiliary cavity modes, simplifying the experimentalimplementation.

Reservoir engineering has earlier been the subject ofsignificant theoretical study in the context of optical andatomic systems [7,20–25], culminating in the experimentaldemonstration by Krauter and co-workers of the entangle-ment of atomic ensembles [26]. The utility of reservoirengineering has also been shown with two-level systems,with demonstrations of superconducting qubit state control[27], as well as entanglement in both trapped ion [28] andsuperconducting [29] systems. In other work pertaining tomechanical entanglement, the entanglement of mechanicalmotion with a microwave field has been demonstrated [30].Earlier, entanglement of phonons at the single-quantum levelwas demonstrated [31], as well as the entanglement of motionaldegrees of freedom of trapped ions [32].

Here, in Sec. II we introduce the multimode optome-chanical system that we shall study. Section III describesapproaches that one may take to reservoir engineering inthis system, while in Sec. IV we describe how these strate-gies could be implemented in our system. In Sec. V, weconsider the adiabatic limit, in which the cavity respondsrapidly to the mechanical motion, and obtain analyticalexpressions for the entanglement, purity, and teleportationfidelity possible with the steady state. Section VI gives ananalysis of the full linearized system, including the effects ofcounter-rotating terms and the possibility of instability. In Sec.VII, we derive a bound on the entanglement based on the cavityoutput spectrum. Section VIII provides an analysis of the three-mode optomechanical system composed of two cavity modescoupled to a single mechanical oscillator, and demonstratesthat the same physics can be realized in this system.

1050-2947/2014/89(6)/063805(17) 063805-1 ©2014 American Physical Society

M. J. WOOLLEY AND A. A. CLERK PHYSICAL REVIEW A 89, 063805 (2014)

FIG. 1. (Color online) (a) The system studied consists of twomechanical oscillators, each coupled to a common cavity (or circuit)mode. (b) Frequencies in this system defined with respect tothe cavity resonance frequency ωc. The blue lines indicate thestandard mechanical sidebands, at ±ωa and ±ωb. If the single-photonoptomechanical coupling rates (ga and gb) are equal, the requiredHamiltonian (8) can be realized using only two cavity drive detunings(±ωm), indicated by the vertical red lines.

II. SYSTEM AND HAMILTONIAN

The system [see Fig. 1(a)] is composed of two mechanicaloscillators, with resonance frequencies ωa and ωb, eachindependently, dispersively coupled (with strengths ga andgb, respectively) to a common cavity mode having resonancefrequency ωc. The Hamiltonian is

H = ωaa†a + ωbb

†b + ωcc†c + ga(a + a†)c†c

+ gb(b + b†)c†c + Hdrive + Hdiss, (1)

where a and b denote mechanical mode lowering operators,c denotes the electromagnetic mode lowering operator, andHdrive accounts for driving of the electromagnetic mode. Theterm Hdiss accounts for dissipation, with the modes subjectto damping at rates γa , γb, and κ , respectively. The systemdynamics, within the usual approximations [33], is describedby the master equation

ρ = −i[H′,ρ] + γa(na + 1)D[a]ρ + γanaD[a†]ρ

+ γb(nb + 1)D[b]ρ + γbnbD[b†]ρ + κD[c]ρ, (2)

with the Hamiltonian H′ = H − Hdiss and the dissipativesuperoperator D[s]ρ = sρs† − 1

2 s†sρ − 12ρs†s.

III. RESERVOIR ENGINEERING STRATEGIES

Given the two mechanical oscillator modes a and b, onecan introduce mechanical two-mode Bogoliubov operators insome rotating frame,

β1 = a cosh r + b† sinh r, (3a)

β2 = b cosh r + a† sinh r. (3b)

Typically, the choice of rotating frame is set by the resonancefrequencies of the system, although here the rotating frameshall be defined with respect to

H0 = (ωa − �)a†a + (ωb + �)b†b + ωcc†c, (4)

with the choice of � to be specified in Sec. IV. Note that inthis frame the collective mechanical quadratures, to be defined

precisely in Eqs. (26a) and (26b), are rotating in a nontrivialmanner. In particular,

X+ = 1

2[ae+i(ωa−�)t + a†e−i(ωa−�)t

+ be+i(ωb+�)t + b†e−i(ωb+�)t ], (5a)

P− = − i

2[ae+i(ωa−�)t − a†e−i(ωa−�)t

+ be+i(ωb+�)t − b†e−i(ωb+�)t ]. (5b)

In any case, the two-mode squeezed state is defined as|r〉2 = S2(r)|0,0〉, where

S2(r) ≡ exp[r(ab − a†b†)] (6)

is the two-mode squeezing operator with squeezing param-eter r [34]. Starting from a |0,0〉 = 0 and b |0,0〉 = 0 itis straightforward to show that [S2(r)aS

†2(r)] |r〉2 = 0 and

that [S2(r)bS†2(r)] |r〉2 = 0. However, β1 = S2(r)aS

†2(r) and

β2 = S2(r)bS†2(r), and therefore, the ground state of β1 and

β2 is the two-mode squeezed state with squeezing parameterr . Our goal then is to engineer the driving Hamiltonian of (1)such that the steady state of (2) results in the βi modes beingcooled to their ground state, implying two-mode squeezing ofthe mechanical oscillators.

One method to achieve this is to use two cavity modesto independently cool the Bogoliubov modes [7–9]. BothBogoliubov modes are independently coupled to a cavitymode with a beam-splitter-like interaction, i.e., β

†i ci + H.c.

(i = 1,2), an interaction which can be used to cool theBogoliubov modes [35]. While such an approach can beeffective, from a practical point of view it would be highlyadvantageous if this could be achieved using only a singlecavity mode.

A seemingly simple way of using only a single reservoirwould be to couple the cavity to one of the Bogoliubov modes,say β1, and then couple β1 to the other Bogoliubov mode(β2) via an all-mechanical contribution to the Hamiltonian ofthe form β

†1β2 + H.c., again a beam-splitter-like interaction.

This interaction will allow β2 to be cooled (by swappingquanta into β1) even though it is not directly coupled to thecooling reservoir. While conceptually simple, the requisitemechanical interaction would be difficult to realize, requiringthe direct coupling of the mechanical oscillators. Such aHamiltonian was considered in Ref. [36], albeit without aphysical implementation specified.

A third approach, which we will pursue here, is to couplethe cavity to the sum of Bogoliubov modes βsum ≡ (β1 +β2)/

√2, and then couple the sum mode to the difference

mode via an all-mechanical Hamiltonian contribution of theform β

†sumβdiff + H.c., where βdiff ≡ (β1 − β2)/

√2. Again,

the swap interaction allows βdiff to be cooled even thoughit is not directly coupled to the cavity. Cooling both βsum

and βdiff is equivalent to cooling both β1 and β2 since〈β†

sumβsum〉 + 〈β†diff βdiff〉 = 〈β†

1β1〉 + 〈β†2β2〉. While this ap-

proach again seems to involve the realization of a challenginginteraction between Bogoliubov modes, this is not the case.The beam-splitter interaction here takes the simple formβ†1β1 − β

†2β2 = a†a − b†b. Thus, one does not require a direct

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TWO-MODE SQUEEZED STATES IN CAVITY . . . PHYSICAL REVIEW A 89, 063805 (2014)

interaction between the mechanical oscillators, but rather justa difference in their resonance frequencies. This is the keyinsight that allows one to appropriately engineer the reservoirvia a single cavity mode.

As noted in Sec. I, we may take another perspective onthis third approach. By introducing a frequency differencebetween the two mechanical oscillators, we are breakingthe degeneracy of the Bogoliubov modes 1 and 2 sincethe Bogoliubov transformation preserves the number-operatordifference. Consequently, the Bogoliubov modes couple todifferent frequency components of the reservoir. Due to thefinite bandwidth of the cavity, it effectively functions as twoindependent reservoirs such that both Bogoliubov modes arecooled.

We thus have that the desired Hamiltonian, in terms of theBogoliubov modes defined in Eqs. (3a) and (3b), is

H = �(β†1β1 − β

†2β2) + G[(β†

1 + β†2)c + H.c.] + Hdiss,

(7)

where � is an effective oscillation frequency and G is aneffective optomechanical coupling. In terms of the originalmechanical annihilation operators, Eq. (7) is

H = �(a†a − b†b) + G+[(a + b)c + H.c.]

+G−[(a + b)c† + H.c.] + Hdiss. (8)

The optomechanical couplings in Eqs. (7) and (8) are relatedby

G ≡√

G2− − G2+, (9a)

tanh r ≡ G+/G−, (9b)

with r being the squeezing parameter entering in thedefinitions of the Bogoliubov modes in Eqs. (3a) and (3b).

Note that if G+ = G− in Eq. (8), then we recover theHamiltonian required for a two-mode backaction-evadingmeasurement of the mechanical oscillators [5], in which twocollective mechanical quadratures commute with the systemHamiltonian. For G+ �= G−, the backaction evasion is lost, butnow there is a backaction that may be regarded as a coherentfeedback process. This enables two-mode squeezing withoutan explicit measurement.

IV. IMPLEMENTATION

The Hamiltonian (8) is readily implemented in conventionalcavity optomechanics setups. We shall focus on the regime|G+| < |G−| such that the dynamics corresponding to (8)are stable. If the single-photon optomechanical coupling ratesin (1) are equal (ga = gb), then just two cavity drives arerequired to realize (8). If they are unequal (ga �= gb), thenfour cavity drives are required. Of course, if ga ∼ gb, we canapproximately realize (8) with only two cavity drives and stillgenerate useful entanglement in the steady state. We considereach of these cases in turn.

A. Two-tone driving

If the single-photon optomechanical coupling rates areequal, we require cavity driving tones at ωc ± ωm, where

ωm = (ωa + ωb)/2 is the average of the two mechanicalfrequencies, i.e.,

Hdrive = (E∗+e+iωmt + E∗

−e−iωmt )e+iωct c + H.c. (10)

This situation is depicted in Fig. 1(b). The driving tonesmust be applied with a fixed relative phase. Working in aninteraction picture defined with respect to the H0 = ωm(a†a +b†b) + ωcc

†c, one finds the effective Hamiltonian to be givenby Eq. (8) where

� = (ωa − ωb)/2, (11)

and the many-photon optomechanical couplings are

G± = (ga + gb)c±/2, (12)

with c± denoting the (assumed real) steady-state amplitudesof the fields at the driven sidebands

c± ≡ 〈c±〉ss = iE±±iωm − κ/2

. (13)

The details of this derivation are given in Appendix A. Itrelies on the assumptions that we are working in the resolved-sideband regime (ωa,ωb κ) and that the driving strengthsE± are large. The former assumption allows us to discardtime-dependent contributions to the Hamiltonian (8), whilethe latter assumption allows us to linearize the optomechanicalinteraction. Note that the ratio G+/G− shall be referred to hereas the drive asymmetry since it is set by the ratio of the cavitydrives on either side of the cavity resonance frequency.

If the single-photon optomechanical couplings are unequal,the two-tone cavity driving can not yield the completematching of oscillator a and b sideband processes requiredin the ideal Hamiltonian of Eq. (8). Instead, there will beadditional contributions to the Hamiltonian (8), given by

Hm = Gm+[(a − b)c + H.c.] − Gm

−[(a† − b†)c + H.c.], (14)

where the effective coupling imperfections are

Gm± = ±(ga − gb)c±/2. (15)

In the two-tone driving case, the imperfection is due to themismatch in the single-photon optomechanical coupling rates.

B. Four-tone driving

The Hamiltonian (8) involves four sideband processes; theup conversion and down conversion of drive photons via theabsorption (or emission) of quanta from (or to) the mechanicaloscillator a or b. The realization of (8) requires a balance of therates at which these processes take place. By using four cavitydriving tones, one tone associated with each sideband process,the balancing of the rates of these processes is possible evenif the single-photon optomechanical couplings are unequal(see Fig. 2). These driving tones are applied with a detuningof � from the mechanical sidebands, at ωc ± (ωa − �) andωc ± (ωb + �), as depicted in Fig. 2. The appropriate Hamil-tonian contribution is

Hdrive = e+iωct c(E∗1+e+i(ωa−�)t + E∗

2+e+i(ωb+�)t

+ E∗1−e−i(ωa−�)t + E∗

2−e−i(ωb+�)t ) + H.c. (16)

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M. J. WOOLLEY AND A. A. CLERK PHYSICAL REVIEW A 89, 063805 (2014)

FIG. 2. (Color online) (a) The three-mode optomechanical sys-tem, as in Fig. 1, under four-tone driving. (b) Frequencies in thissystem defined with respect to the cavity resonance frequency ωc.The blue lines indicate the standard mechanical sidebands, at ±ωa and±ωb. If the single-photon optomechanical coupling rates are unequal,the required Hamiltonian (8) can be realized using four cavity drivingfrequencies ±(ωa − �), ± (ωb + �), indicated by vertical red lines.

The steady-state amplitudes at the driven sidebands aredenoted by ck± (k = 1,2), with

ck± ≡ 〈ck±〉ss = iEk±±iωk − κ/2

, (17)

where we have introduced the notation for the drive detunings

ω1 ≡ ωa − �, (18a)

ω2 ≡ ωb + �. (18b)

Then, we demand that the driving strengths are “matched,”meaning that

c1±c2±

= gb

ga

. (19)

That is, we require that the two steady-state amplitudes (i.e.,drives) on the same side of the cavity resonance frequency havean asymmetry set by the optomechanical coupling asymmetry.With the condition (19) satisfied, and working in an interactionpicture defined with respect to the Hamiltonian (4), the systemcan again be described by the Hamiltonian (8), now with the(assumed real) many-photon optomechanical coupling rates

G± = (gac1± + gbc2±) /2. (20)

Again, the details of the derivation are left to Appendix A.Imprecision in the matching condition (19) gives additionalcontributions to the Hamiltonian (8), of the form of Eq. (14),but now with

Gm± = ± (gac1± − gbc2±) /2. (21)

In this case, the effective coupling imperfection arises fromthe drives not being weighted precisely according to thecondition (19).

The cavity drive frequencies should be set such that �

satisfies the following conditions:

� γ, (22a)

� � (ωa − ωb)/2 − γ. (22b)

Condition (22a) ensures that the sum and difference Bo-goliubov modes are sufficiently coupled [cf. Eq. (7)]. The

condition (22b) ensures that the unwanted sideband processeshave a negligible effect on the system dynamics. It issometimes convenient to refer to the drive frequencies via theirdetunings from the center of the two mechanical sidebands,given by

δ ≡ (ωa − ωb)/2 − �. (23)

It is interesting to note that with the driving conditionc1+ = c1−, in addition to (19), we could realize a two-modebackaction-evading measurement of the mechanical oscillatorsirrespective of the coupling asymmetry, and so generalize theresults of Ref. [5].

V. ADIABATIC LIMIT

We first consider the dynamics of the system governed by(8) in the adiabatic limit, where the cavity responds rapidlyto the mechanical motion; that is, where κ > �,G± (but stillin the regime where ωa,ωb κ). In this limit, we eliminatethe cavity mode, obtaining an effective description for themechanical modes alone. This adiabatic limit will simplify theanalysis and thus provide insight into our mechanism; it willalso prove to be a useful regime for the task of mechanicalentanglement generation.

We stress that the Hamiltonian (8) applies both to thecase of equal single-photon optomechanical couplings andtwo-tone driving, and to the case of unequal single-photonoptomechanical couplings with matched four-tone driving(although � is determined differently in each case). Further,the imperfections (asymmetry in couplings in the former caseand mismatch in driving conditions in the latter case) are bothdescribed by the Hamiltonian (14).

In this adiabatic limit, the cavity annihilation operatoris given by c = −2iG(β1 + β2)/κ . Substituting this into thedissipative terms of the master equation (2), the adiabaticallyeliminated master equation is

ρ =−i�[β†1β1 − β

†2β2,ρ] + γa(na + 1)D[a]ρ + γanaD[a†]ρ

+ γb (nb + 1)D[b]ρ + γbnbD[b†]ρ + D[β1 + β2]ρ,

(24)

with the optomechanical damping rate

≡ γ C ≡ 4G2

κ, (25)

whereG is the effective optomechanical coupling introduced inEq. (9a), and C is the corresponding cooperativity parameter.Now, the steady state of Eq. (24) is easily obtained, and itsentanglement and purity metrics readily calculated.

In view of Eq. (24), an alternative interpretation of thecooling of both Bogoliubov modes is possible. In the limitγ /� → 0, the terms containing β

†1β2 and β

†2β1 will rapidly

average away, meaning that Eq. (24) will be equivalent tohaving independent dissipation of modes β1 and β2. Physically,this corresponds to the dissipation of β1 and β2 being due todistinct modes of the reservoir.

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TWO-MODE SQUEEZED STATES IN CAVITY . . . PHYSICAL REVIEW A 89, 063805 (2014)

A. Entanglement

The case of symmetric mechanical damping (γa,γb = γ )and symmetric thermal occupation of the mechanical baths(na,nb = n) allows simple analytical results for the steady-state second moments to be obtained. The assumption ofequal thermal occupations is reasonable for most experimentalsituations, while it turns out that our results are not sensitiveto unequal mechanical damping rates provided that theyare both small. The simplest two-mode, continuous-variableentanglement criterion is provided by the Duan inequality [37].It is expressed in terms of collective quadrature operators,defined by

X± = (Xa ± Xb)/√

2, (26a)

P± = (Pa ± Pb)/√

2, (26b)

where we have introduced the usual quadratures for eachoscillator mode

Xs = (s + s†)/√

2, Ps = −i(s − s†)/√

2. (27)

Then, the Duan criterion tells us that a Gaussian state for which

〈X2+〉 + 〈P 2

−〉 < 1 (28)

is inseparable. Note that this could equally well be formulatedin terms of X− and P+, although Eq. (28) shall be the suitableform here.

For our system we consider the limit γ /� → 0 since thisensures that the sum and difference Bogoliubov modes aresufficiently coupled (or, equivalently, that the two individualBogoliubov modes see effectively independent reservoirs). ForG− �= G+, we find for the steady-state second moments

〈X2±〉 = 〈P 2

∓〉 = γ

γ + (n + 1/2) +

γ +

e∓2r

2(29a)

= γ κ

γ κ + 4(G2− − G2+)(n + 1/2)

+ 2(G− ∓ G+)2

γ κ + 4(G2− − G2+). (29b)

Equation (29a) takes a particularly simple form, describingcoupling to a squeezed reservoir with an optomechanicaldamping rate . The results (29a) and (29b) are easily checkedagainst the solution of the full system (i.e., without theadiabatic elimination), as discussed in Sec. VI.

The G+ = G− limit is unclear from Eq. (29a), as itcorresponds to the limits → 0 and r → +∞. However,the result is clear from Eq. (29b), and we recover the resultthat 〈X2

+〉 = 〈P 2−〉 = n + 1/2 (evading the backaction) and

〈X2−〉 = 〈P 2

+〉 = n + 1/2 + C±/2 (heated by the backaction)[5], where the cooperativities associated with the blue and redsideband drives alone (denoted by the subscripts “+” and “−”,respectively), are

C± ≡ ±γ

≡ 4G2±

γ κ. (30)

While the Duan inequality provides a simple entanglementcriterion, the entanglement may be quantified via the loga-rithmic negativity, defined in Appendix B 1. Both are shown,as functions of the drive asymmetry G+/G−, in Fig. 3. In

Fig. 3(a), it is seen that the Duan quantity takes a value belowone for experimentally reasonable parameters, achievable instate-of-the-art microwave cavity optomechanics experiments[12–14], indicating that the mechanical oscillators are entan-gled in the steady state. This continues to be the case even whenone accounts for large nonzero initial thermal occupationsand large imperfections in the effective couplings. Further,the logarithmic negativity, shown in Fig. 3(b), takes a largevalue for these parameters. For comparison, the logarithmicnegativity of a stationary two-mode squeezed state generatedvia a Hamiltonian parametric amplifier interaction is boundedabove, due to a stability constraint, by EN = ln 2 ∼ 0.69.

As previously noted [8,17], the entanglement goes througha maximum as a function of the drive asymmetry. For themechanical steady state to be highly entangled, we requireboth that the target steady state is highly squeezed (r large,requiring G+/G− → 1) and that the system is effectivelycooled towards this steady state ( large, requiring G+/G− →0). Obviously, these limits are incompatible and the optimalasymmetry is between these, leading to the observed maxi-mum.

Given the simple analytical results we have obtained, wemay optimize the steady-state entanglement analytically byminimizing the Duan quantity over the drive asymmetry. Thisis most conveniently done using Eq. (29b). We find that theoptimal drive asymmetry is

G+G−

∣∣∣∣opt.

= 1 + 1 + n

C−−

√1 + 1/C−

C−(31a)

∼ 1 − 1√C−

. (31b)

The result (31b) holds in the large-cooperativity limit,provided that one is still within the adiabatic regime. It followsthat the Duan quantity (28), to first order in C−1

− , is

〈X2+〉 + 〈P 2

−〉 = 1 + n√C−

+ n(1 + n)

C−. (32)

Clearly, the mechanical oscillators are entangled even for amodest cooperativity. We emphasize these results are onlyvalid in the adiabatic limit, corresponding to C− � 4κ/γ . Theachievable entanglement beyond the adiabatic regime shall bediscussed in Sec. VI.

B. Purity

The purity of the steady state generated is relevant forboth experiments in quantum foundations and in quantuminformation processing; its role in determining teleportationfidelity shall be described in Sec. V C. Now, the fact that thesteady state is highly entangled does not necessarily imply thatthe steady state is highly pure. Indeed, if one cools only oneBogoliubov mode, then the steady state is highly entangled, butalso highly impure [8]. The purity of the mechanical two-modesteady state is defined as

μ ≡ tr(ρ2), (33)

where ρ is the density matrix of the two mechanical modes. Itcan be directly evaluated from knowledge of its symmetricallyordered covariance matrix V. With quadratures as defined in

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M. J. WOOLLEY AND A. A. CLERK PHYSICAL REVIEW A 89, 063805 (2014)

FIG. 3. (Color online) Entanglement, expressed via (a) the Duanquantity (28) and (b) the logarithmic negativity, against the driveasymmetry G+/G−. We hold G− fixed (fixing the cooperativityC−) and vary G+. These results correspond to the time-independent(rotating-wave approximation) Hamiltonian (8), and apply to both thetwo-tone and four-tone driving cases, with the effective couplings G±given by Eqs. (12) and (20), respectively. These plots are obtainedusing the adiabatic limit results of Sec. V, although they coincide withthe results for the full system obtained in Sec. VI A. The solid blackcurve corresponds to a mechanical bath thermal occupation of n = 0,and Gm

± = 0 where Gm± are the effective coupling imperfections,

introduced in Eqs. (14) and (21) for the two-tone and four-tonedriving cases, respectively. The blue curve (long dashes) correspondsto n = 25 and Gm

± = 0, while the red curve (short dashes) correspondsto n = 25 and Gm

± = 0.5G±. Parameters common to each curve areC− = 1200, κ = 2π × 1.592 × 105 Hz = 106 s−1, γ /κ = 4 × 10−5,and �/κ = 0.1.

Eqs. (27) and the covariance matrix expressed in the orderedbasis (Xa,Pa,Xb,Pb), the purity is simply given by

μ = 1/(4√

det V). (34)

The purity may also be assessed by calculating the thermaloccupations of the two mechanical Bogoliubov modes, definedin Eqs. (3a) and (3b).

Plots of both quantities, against the drive asymmetryG+/G−, are shown in Fig. 4. From Fig. 4(a) it is clear thatthe occupations of the two Bogoliubov modes are the sameprovided that the imperfections in the effective couplings,Eqs. (15) and (21), are zero. Further, the occupations are closeto zero for reasonable parameters, verifying that our schemeeffectively cools both Bogoliubov modes. The purity of thestate is correspondingly high, being close to one for reasonableexperimental parameters [see Fig. 4(b)], and so outperforminga scheme in which only one Bogoliubov mode is cooled [8].

In the absence of effective coupling imperfections, we canobtain simple analytical results characterizing the purity of thesteady state. The occupations of the Bogoliubov modes in the

FIG. 4. (Color online) (a) Steady-state occupations of the me-chanical Bogoliubov modes, defined in Eqs. (3a) and (3b), and (b)steady-state purity of the two mechanical modes, defined in Eq. (33),against drive asymmetry. The solid black curve corresponds to amechanical bath thermal occupation of n = 0 and no imperfectionin the effective couplings (Gm

± = 0), the blue curve (long dashes)corresponds to n = 25 and Gm

± = 0, and the red curves (short dashes)correspond to n = 25 and Gm

± = 0.5G±. Only one solid black curveand one dashed blue curve is shown in (a) since the occupations ofthe two Bogoliubov modes are the same in each of these cases. Theblack curves with long and short dashes correspond to bounds onthese quantities for the scheme in which only one Bogoliubov modeis cooled [8], assuming a thermal occupation of n = 0: the curve in(a) is a lower bound on the occupation of the uncooled Bogoliubovmode, while the curve in (b) is an upper bound on the purity of thesteady state in this case. Remaining parameters for each curve are asgiven in the caption of Fig. 3.

limit γ /� → 0 (and for G− �= G+) are

〈β†i βi〉 = γ

γ + [n + (2n + 1) sinh2 r] (35a)

= γ κ

γ κ + 4(G2− − G2+)

[G2

+ + n(G2− + G2

+)

G2− − G2+

], (35b)

for i = 1 and 2, consistent with the result for single-modesqueezing in Ref. [17]. Again, it is clear that both Bogoliubovmodes are cooled equally. The purity itself, in the limit γ /� →0, is given by

μ = (γ + )2

[γ (1 + 2n) + ]2 + 4(1 + 2n)γ sinh2 r. (36)

The purity at the entanglement maximum, in the large-cooperativity limit, is μ ∼ 1/[2(1 + n)]. However, a highlypure (and still highly entangled) steady state can be achievedby choosing a drive asymmetry just below that correspondingto the optimal entanglement (as this gives a large and hencemore cooling, at the expense of a smaller squeeze parameter r).

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C. Teleportation fidelity

The two-mode squeezed state generated here may beregarded as the entangled resource (“EPR channel”) in acontinuous-variable teleportation protocol [18]. If we writethe steady-state two-mode symmetrically ordered covariancematrix, in the ordered basis (Xa,Pa,Xb,Pb), in block form as

V =[

Va Vab

VTab Vb

], (37)

then the teleportation fidelity for a single-mode Gaussian inputstate under the standard protocol is given by [38]

F = 1√det(2Vin + N)

, (38a)

N = σzVaσz + σzVab + VTabσz + Vb, (38b)

where Vin is the covariance matrix of the state to be teleported.For the teleportation of a coherent state, this is Vin = (1/2)I2.

We find that the mechanical steady state in our system, inthe limit γ /� → 0, is a thermal two-mode squeezed state.Such a state is defined by

ρ ≡ S2(ξ )(ρa

th ⊗ ρbth

)S†2(ξ ), (39)

where ρa(b)th denotes the density matrix of a thermal state of

mode a(b) with occupation na(b)th , and S2(ξ ) is the two-mode

squeezing operator introduced in Eq. (6). Accordingly, wecan assign effective occupations n

a(b)th and an effective two-

mode squeezing parameter ξ to our steady state, as detailed inAppendix B 2. Now, the purity of such a state is simply

μ = 1(1 + 2na

th

)(1 + 2nb

th

) , (40)

and the teleportation fidelity, based on a thermal two-modesqueezed state channel, is [38]

F = 1

e−2ξ(1 + na

th + nbth + e2ξ

) . (41)

Clearly, larger effective occupations correspond to a lowerpurity and a lower teleportation fidelity.

It is known that when the channel of a continuous-variableteleportation protocol is a symmetric thermal two-modesqueezed state (na

th,nbth ≡ nth), the teleportation fidelity is

simply [19]

F = 1

1 + e−EN, (42)

where the logarithmic negativity is given by EN =Max[0,2ξ − ln(1 + 2nth)]. Equation (42) actually gives theoptimal teleportation fidelity achievable for a given amountof entanglement [19]. Asymmetry results in a teleportationfidelity below that given by Eq. (42).

The effective thermal occupations for our steady state(na

th and nbth), as a function of the drive asymmetry, are

shown in Fig. 5(a). It is clear that nath = nb

th provided that theimperfections in the effective couplings are zero. Accordinglyin these cases, the teleportation fidelity is given by Eq. (42) andso attains its optimal value for a given amount of entanglement.

Consider now the case of an asymmetric channel ntha �= nth

b ;this is generically the kind of state produced using a schemewhich cools only a single Bogoliubov mode [8]. For such

FIG. 5. (Color online) (a) Effective thermal occupations for eachmechanical mode in the steady state, introduced in Eq. (39), againstthe drive asymmetry. (b) The teleportation fidelity, using the generatedmechanical steady state as an EPR channel to teleport a coherent statevia the standard protocol, against the drive asymmetry. The solid blackcurves correspond to a mechanical bath thermal occupation of n = 0and no imperfection in the effective couplings (Gm

± = 0), the bluecurves (long dashes) correspond to n = 25 and Gm

± = 0, and the redcurves (short dashes) correspond to n = 25 and Gm

± = 0.5G±. Theblack curve with long and short dashes on the left is the lower boundon the occupation of the uncoupled mode in a scheme that cools oneBogoliubov mode [8]. The black curve with long and short dashes onthe right is the teleportation fidelity achievable with such a scheme.The upper (lower) dotted purple curve is the upper (lower) bound onthe optimal teleportation fidelity achievable for a given amount ofentanglement. The given amount of entanglement corresponds to thatpossessed by a two-mode squeezed state with squeezing parameterr = tanh−1 (G+/G−) [cf. Eq. (9b)]. Remaining parameters for eachsolid curve are as given in the caption of Fig. 3.

states, the standard teleportation protocol does not achieve thefidelity in Eq. (42). This fidelity can be reached in principlein the highly entangled regime, if one goes beyond thestandard protocol by allowing for additional local Gaussianoperations [39].

The teleportation fidelity, assuming that the mechanicalsteady state we have generated is used as an EPR channeland assuming no other sources of imperfection, is plottedas a function of drive asymmetry in Fig. 5(b). Crucially, bycooling both Bogoliubov modes, the optimal teleportationfidelity tends to 1 rather than to 4

5 in the highly entangledregime. Further, this upper bound is achievable with reasonableparameters.

D. Coherent feedback

The reservoir engineering scheme that we have describedhere permits an alternative interpretation in terms of coherentfeedback [16], similar to that provided for the squeezingscheme in Ref. [17]; we depict the process schematically inFig. 6. The Hamiltonian (8) can be rewritten in terms of the

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FIG. 6. (Color online) Representation of the reservoir engineer-ing scheme of Figs. 1 and 2 as a form of coherent feedback, with thefeedback (backaction) being applied autonomously via the cavitymode. The description is in terms of the collective mechanicalquadrature operators introduced in Eqs. (26a) and (26b). (a) Thereis a measurement of X+ via the cavity at a rate ∼(G− + G+) anda feedback onto X+ via the cavity at a rate ∼(G− − G+). Themeasurement rate being greater than the feedback rate leads tosqueezing. (b) There is a measurement of P+ via the cavity at arate ∼(G− − G+) and a feedback onto P+ via the cavity at a rate∼(G− + G+). The feedback rate being greater than the measurementrate leads to amplification.

collective mechanical quadratures of Eqs. (26a) and (26b) as

H = �(X+X− + P+P−) +√

2(G− + G+)X+Xc

+√

2(G− − G+)P+Pc + Hdiss. (43)

This Hamiltonian is a perturbation, via the third term, of aHamiltonian that we previously studied in the context of two-mode backaction-evading measurement and feedback control[5]. From a coherent feedback point of view, the second term inEq. (43) may be regarded as the “measurement” interaction andthe third term in Eq. (43) may be regarded as the “feedback”backaction, applied autonomously via the cavity mode.

After an adiabatic elimination of the cavity mode, theHeisenberg-Langevin equations corresponding to the Hamil-tonian (43) may be written as

d

dt�X = A0 · �X + B1 · �Xin + B2 · �Yin, (44)

where �X = (X+,P+,X−,P−)T is the vector of mechanicalcollective quadrature observables, �Xin is the correspondingvector of mechanical input noises, and �Yin ≡ [Y1(t),Y2(t)]T

are the operators associated with the cavity input noise. Theremaining matrices in Eq. (44) are specified in Appendix C 1.The new noise input operators have the correlation functions

〈Y1(t)Y1(0)〉 = 1

2

G− − G+G− + G+

δ(t) ≡(

n1 + 1

2

)δ(t), (45a)

〈Y2(t)Y2(0)〉 = 1

2

G− + G+G− − G+

δ(t) ≡(

n2 + 1

2

)δ(t), (45b)

where n1 and n2 denote effective thermal occupations ofthe noise inputs. Clearly, the input noises seen by thecollective mechanical quadratures are weighted by the ratiosof the measurement and feedback rates. Since we haveG− > G+ � 0 here, the effective occupation n1 is negative[17]. Therefore, as far as the collective quadrature X+ isconcerned, the cavity behaves as a squeezed bath and thereforeX+ will be squeezed in the steady state [33]. Conversely, n2 is

positive and the collective quadrature X− will be antisqueezedin the steady state.

VI. FULL SYSTEM

A. Solution with time-independent Hamiltonian

In Sec. V, we calculated the steady state of our system inthe adiabatic limit, after mathematically removing the cavitymode from the system. Even with the cavity mode retained, theeffective Hamiltonian (8) is quadratic and time independent,and we may readily solve for the steady state. The Heisenberg-Langevin equations corresponding to (8) may be written as thesystem

d

dt�X = A0 · �X + B0 · �Xin, (46)

where �X = (Xa,Pa,Xb,Pb,Xc,Pc)T is defined in terms ofindividual oscillator quadratures, and the matrices are specifiedin Appendix C 2. The steady-state, symmetrically orderedcovariance matrix V is obtained by solving the Lyapunovequation

A0V + VAT0 = −B0BT

0 . (47)

Solving Eq. (47) allows us to assess the steady state even whenwe are not in the adiabatic limit, and to validate results obtainedin the adiabatic limit. As before, knowledge of the covariancematrix allows the evaluation of entanglement, purity, andfidelity measures. The analytical results are easily obtainedbut sufficiently complicated that we do not quote them here,while the numerical results coincide with those previouslyobtained in the adiabatic limit.

B. Solution with time-dependent Hamiltonian

The results of Sec. VI A are still only valid provided thatwe are justified in making a rotating-wave approximation;that is, in discarding the time-dependent (“counter-rotating”)contributions that arise in the derivation of the Hamiltonian (8).Here, we account for these time-dependent contributions;the exact forms that they take are given in Appendix D 1.The corresponding contributions to the Heisenberg-Langevinequations can be handled by making the replacement A0 →A(t) in Eq. (46), where the time-dependent drift matrix is givenby

A(t) = A0 +N∑

k=1

(Ak+e+2iδk t + Ak−e−2iδk t ). (48)

For the case of two-tone driving (10), N = 1 and δ1 = ωm.For four-tone driving (16), we have N = 4 and δ1 = δ, δ2 =ωb + �, δ3 = ωm, and δ4 = ωa − � [see Fig. 2 and recall thatδ was introduced in Eq. (23)]. The first of these contributionsis due to the second drive on the same side of the cavityresonance frequency, while the remaining contributions aredue to the drives on the other side of the cavity resonancefrequency. The matrices Ak± are given in Appendix D 2.

Given that the drift matrix (48) is now time varying, thecovariance matrix V(t) is given by solution of the Lyapunov-like differential equation

V = AV + VA† + B0BT0 . (49)

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FIG. 7. (Color online) The effects of counter-rotating contributions to the Hamiltonian, described in detail in Appendix D, on theentanglement and purity of the mechanical two-mode steady state under four-tone driving. The plots show (a) the Duan quantity, (b) thelogarithmic negativity, (c) the occupations of the Bogoliubov modes, and (d) the purity, as functions of the drive asymmetry. The solid blackcurve corresponds to the time-independent Hamiltonian (8), while the other curves correspond to cases where counter-rotating effects aresignificant. The blue curve (long dashes) is for δ/κ = 0.5, the red curve (short dashes) is for ω1/κ = 5, and the purple curve (long and shortdashes) is for ga/gb = 1.5. Although it is possible for counter-rotating contributions to have a significant effect on these measures, it should alsobe possible to achieve sufficiently high sideband resolution that their effects may be neglected. Note that under two-tone driving we typicallyhave ga = gb and always have δ = 0; the effect of the parameter ωm/κ in that case is comparable to the effect of the parameter ω1/κ in thefour-tone driving case. Parameters, unless otherwise specified, for each curve are δ/κ = 1, ω1/κ = 100, and d = 1. The parameters commonto each curve are n = 0, Gm

± = 0, C− = 1200, κ = 2π × 1.592 × 105 Hz = 106 s−1, γ /κ = 4 × 10−5, and �/κ = 0.1.

The covariance matrix will be oscillatory in the long-timelimit; we seek the dc component of its solution. The directnumerical solution of Eq. (49) is inefficient, so instead we usean ansatz to obtain an approximate numerical solution [40].The procedure used is outlined in Appendix D 3.

C. Effects of counter-rotating terms

The effects of the counter-rotating Hamiltonian contribu-tions on the entanglement and purity of the steady state areshown in Figs. 7 and 8. Figure 7 shows these as functions ofdrive asymmetry, while Fig. 8 shows the two-mode squeezing(entanglement), optimized over the drive asymmetry, as afunction of the cooperativity parameter C−.

From Fig. 7 it is clear that there is a degradation in theentanglement and purity measures as the frequency of thecounter-rotating terms is lowered. This is unsurprising sincethe form of the time-dependent Hamiltonian contributions,detailed in Appendix D 1, depart from the ideal form of Eq. (7).However, the same overall behavior in the entanglement andpurity is observed; that is, a maximum in the entanglement anda monotonic decrease in the purity.

The rotating-wave approximation results coincide withthe full time-dependent Hamiltonian results in the limit thatall counter-rotating frequencies greatly exceed the cavitydecay rate; that is, all |δk| κ . With the parameters chosen,counter-rotating effects become significant at δ/κ ∼ 0.5 andωa/κ ∼ 5; these correspond to modest sideband resolutions.

FIG. 8. (Color online) Mechanical two-mode squeezing as afunction of the cooperativity parameter C−, introduced in Eq. (30),for a range of sideband resolutions ωm/κ . The results are presentedfor the case of two-tone cavity driving with the single-photonoptomechanical coupling rates being equal, ga = gb (that is, no im-perfection in the effective coupling rates). The quantity plotted is thetwo-mode squeezing in dB, defined by TMS (dB) ≡ − log10[(〈X2

+〉 +〈P2

−〉)/(〈X2+〉 + 〈P2

−〉)0], with both the numerator and denominatorbeing instances of the Duan quantity of Eq. (28). At each value ofthe cooperativity parameter C−, the Duan quantity is minimized overthe effective coupling asymmetry G+/G−. The curves are shown fora rotating-wave approximation (solid black curve) meaning that thesideband resolution is effectively infinite; ωm/κ = 103 (blue curvewith long dashes); ωm/κ = 102 (purple curve with short dashes);ωm/κ = 10 (red curve with long and short dashes). In the case offour-tone cavity driving, the behavior of the two-mode squeezingas a function of ω1/κ is similar to the behavior seen here as afunction of ωm/κ . The other parameters are as specified in the captionof Fig. 7.

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M. J. WOOLLEY AND A. A. CLERK PHYSICAL REVIEW A 89, 063805 (2014)

It should be possible to significantly exceed this resolutionand therefore largely avoid the effects of counter-rotatingterms. The ratio δ/κ can be reduced further than the ratioωa/κ without significant deleterious effects due to the distinctmanner with which the corresponding contributions enter thefull time-dependent Hamiltonian. Provided that the asymmetryin the single-photon optomechanical coupling rates is small,the effective couplings associated with the terms rotating at±2δ are not exponentially enhanced in the large-r limit, whilethose oscillating at ±2ωa , ±2ωm, and ±2ωb are exponentiallyenhanced [see Eq. (D3)].

Note that with counter-rotating terms included, the entan-glement and purity of the mechanical steady state depend onthe asymmetry in the single-photon optomechanical couplingrates, even if there are no imperfections in the effectivecouplings. This is in contrast to the results obtained withthe time-independent Hamiltonian (8). As seen in Fig. 7, thisasymmetry leads to a significant degradation in entanglementand purity measures if the imperfection is around ∼50% ofthe effective coupling. Again, it should be possible to engineerthe optomechanical system such that the asymmetry is muchlower than this value, and the corresponding deleterious effectsare negligible.

The behavior of the optimized mechanical two-modesqueezing as a function of the cooperativity parameter C− isshown in Fig. 8. Even neglecting the effects of counter-rotatingterms, the achievable two-mode squeezing (entanglement)plateaus in the large-cooperativity limit. For a finite sidebandresolution, however, the squeezing goes through a maximum asa function of the cooperativity, with the maximum occurringat a lower value of the cooperativity parameter as the side-band resolution is decreased. Unsurprisingly, the discrepancybetween the rotating-wave approximation (RWA) result andthe full result increases as the sideband resolution is reduced.Note, however, that these discrepancies become significantonly at very large values of the cooperativity parameter.Also note that in the large-cooperativity limit, it is possiblefor the dynamics associated with the full time-dependentHamiltonian to be unstable where the dynamics associatedwith the corresponding time-independent Hamiltonian arestable. However, this does not tend to be the case at theoptimal drive asymmetry. This potential for the onset of aninstability occurs at higher values of the cooperativity thanwe have previously considered in this work. Our results showthat for high levels of steady-state entanglement, the reservoirengineering scheme discussed here is robust against realisticlevels of counter-rotating corrections.

VII. EXPERIMENTAL OBSERVABILITY

A. Output spectrum

From an experimental point of view, reconstructing theentire covariance matrix would be extremely demanding. Evenperforming direct measurements of both of the collectivequadatures required for testing the Duan criterion of Eq. (28)would be difficult. However, one could perform a backaction-evading measurement of the collective quadrature X+, and takethis as some evidence for the existence of two-mode squeezingin the steady state. Alternatively, we can seek a signature ofthe mechanical entanglement in the cavity output spectrum.

As usual, the output spectrum is calculated by first solvingthe Heisenberg-Langevin equations in the frequency domain.Taking the Fourier transform of Eq. (46) we find

�X[ω] = −(A0 + iωI6)−1 · B0 · �X[ω], (50)

where �X[ω] = (a[ω],a†[ω],b[ω],b†[ω],c[ω],c†[ω])T and thematrices are given in Appendix C 3. The output spectrum iscalculated in the standard manner [33] as

S[ω] =∫

dt eiωt 〈δc†out(t)δcout(0)〉, (51)

where the output cavity field is given by δcout = cout − 〈cout〉and cout = cin + √

κcin.We first calculate the spectrum assuming that there are

no imperfections in the effective couplings (Gm± = 0) and

ignoring time-dependent Hamiltonian contributions; that is,with the effective Hamiltonian (8). Then, the cavity outputspectrum, in the limit γ /� → 0, is given by

S[ω] = κ32[G2

−n + G2+(n + 1)]γ [γ 2 + 4(ω2 + �2)]

|N (ω)|2 , (52)

where N (ω) = [8G2 + (γ − 2iω)(κ − 2iω)](γ − 2iω) +4�2(κ − 2iω). In the case � = 0, this reduces to the resultfor a single mechanical oscillator [17]. As shown in Fig. 9, theoutput spectrum exhibits peaks at detunings around ±� fromthe cavity resonance frequency. This corresponds to the drivephotons being scattered towards the cavity resonance, withan energy ωa or ωb being provided by or extracted from themechanical oscillators. As the optomechanical damping rate

is increased, the widths of the spectral peaks increase and theyare shifted to larger detunings (for G−/G+ > 0), as expected.

If we now allow for the possibility of imperfections in theeffective couplings, but still ignore time-dependent contribu-tions, the cavity output spectrum is again readily obtained. Thecorresponding spectra are also shown in Fig. 9. The generalexpression is complicated, but at detunings of ±� we find

S[±�] = γ κ(G− ± Gm

−)2n + (G+ ± Gm+)2(1 + n)

[G2− − (Gm−)2 − G2+ + (Gm+)2]2. (53)

Clearly, the asymmetry in the spectral peaks is determinedby the imperfections Gm

± in the effective optomechanicalcouplings.

B. Bogoliubov modes

Knowledge of the cavity output spectrum can be used toprovide us with information about the occupations of themechanical Bogoliubov modes. In particular, neglecting im-perfections in the effective couplings and in the limit γ /� →0, we can show that the occupations of the Bogoliubov modesare related to the integral of each peak in the output spectrum by∫ 0

−∞S[ω]dω =

∫ +∞

0S[ω]dω

= 8πκG2

4G2 + κ(κ + γ )〈β†

i βi〉, (54)

for i = 1 or 2 (in this, and subsequent, expressions). Therefore,from the output spectrum and knowledge of the system param-eters, one can determine the occupations of the Bogoliubov

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FIG. 9. (Color online) Cavity output spectra, as defined in Eq. (51), centered around detunings from the cavity resonance frequency of (leftpanel) ω = −� and (right panel) ω = +�. The spectra are normalized to the peak in the output spectrum in the absence of imperfections in theeffective couplings, denoted by S0[ω]|max. The spectra are shown for the case (solid black curve) without imperfections in the effective couplings(Gm

± = 0), and for the cases Gm±/G± = 0.3 (blue curve with long dashes) and Gm

±/G± = −0.3 (red curve with short dashes). Imperfections inthe effective couplings lead to asymmetry in the observed output spectra [see Eq. (53)]. In the absence of these imperfections, the steady-statemechanical entanglement can be bounded based on a measurement of the output spectrum. The spectra are shown for a drive asymmetryG+/G− = 0.9 and a cooperativity C− = 1200, while other parameters are n = 0, κ = 2π × 1.592 × 105 Hz = 106 s−1, γ /κ = 4 × 10−5, and�/κ = 0.1.

modes. The same information can be obtained from the heightsof the spectral peaks since (in the same limit) we also have

S[±�] = γ κ + 4(G2− − G2

+)

G2− − G2+〈β†

i βi〉. (55)

Note that Eqs. (54) and (55) only hold when the imperfectionsin the effective couplings are less than ∼ 1%. A similar resultto that of Eq. (54) was obtained for a single mechanicaloscillator [17], although in that case the integration is over allfrequencies.

C. Entanglement criterion

Now, from Eqs. (54) and (55) it is clear that we can estimatethe occupations of the Bogoliubov modes using the cavityoutput spectrum. Recall that the simplest means of verifyingthe presence of mechanical entanglement is via the Duancriterion (28). The task then is to bound the Duan quantityusing our knowledge of the occupations of the Bogoliubovmodes.

Repeated application of the generalized Cauchy-Schwarzinequality [17,41] allows one to show that |〈β2

i 〉| � 〈β†i βi〉 +

12 . With the additional assumption that 〈β†

1β1〉 = 〈β†2β2〉,

known to be true in the absence of imperfections in thecouplings and in the limit γ /� → 0, we can bound the Duanquantity. Explicitly, we find that

〈X2+〉 + 〈P 2

−〉 � 8e−2r (〈β†i βi〉 + 1/2). (56)

The parameter r is known from the drive asymmetry [cf.Eq. (9b)]. The Duan quantity and its bound converge in thehighly entangled (large-r) regime, and therefore we expect itto reliably indicate the existence of an entangled mechanicalsteady state.

VIII. TWO CAVITY MODES, ONE MECHANICALOSCILLATOR

Thus far, we have considered a three-mode optomechanicalsystem composed of two mechanical oscillators coupled to onecommon cavity mode. It is of considerable interest, particularly

from the perspective of quantum information processing, toconsider the opposite scenario in which there are two cavitymodes coupled to a single mechanical oscillator [42–45]. Inparticular, the ubiquity of optomechanical couplings raisesthe possibility of entangling cavity modes of vastly differentfrequencies (e.g., microwave and optical modes). This three-mode optomechanical system is depicted schematically inFig. 10(a).

Now, the Hamiltonian of the system is [cf. Eq. (1)]

H = ωaa†a + ωbb

†b + ωcc†c + ga(c + c†)a†a

+ gb(c + c†)b†b + Hdrive + Hdiss, (57)

where a and b describe two cavity modes, and c describes themechanical oscillator. We shall assume a driving Hamiltonianof the form

Hdrive = (Ea+e+iωct + Ea−e−iωct )e+i(ωa−�)t a

+ (Eb+e+iωct + Eb−e−iωct )e+i(ωb+�)t b + H.c. (58)

FIG. 10. (Color online) (a) Representation of a three-mode op-tomechanical system composed of two cavity modes (with resonancefrequencies ωa and ωb), each independently coupled by radiationpressure to a single mechanical oscillator (with resonance frequencyωc). (b) The driving conditions, in terms of frequencies, that lead to thelinearized effective Hamiltonian (8) with the role of the cavity andmechanical modes interchanged. The cavity resonance frequenciesare indicated by the blue lines, while the driving tones are indicatedby red lines. The driving tones are placed symmetrically (at detunings±ωc) about detunings of � from the cavity resonance frequencies.

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M. J. WOOLLEY AND A. A. CLERK PHYSICAL REVIEW A 89, 063805 (2014)

That is, driving at ωa − � ± ωc and ωb + � ± ωc for cavitymodes a and b, respectively, as depicted in Fig. 10(b). Inan interaction picture defined with respect to the Hamiltonian(4), the Hamiltonian (57) takes the form of the Hamiltonian (8)if we set gaa+ = gbb+ ≡ G+ and gaa− = gbb− ≡ G−, witha± and b± denoting the steady-state amplitudes at the drivencavity sidebands. Thus, we can realize the same physics withthe cavity modes that we described previously for the me-chanical modes, including the possibility of generating highlypure, highly entangled electromagnetic modes. As before, thisderivation relies on the assumptions that we are operatingin the resolved-sideband regime, that the driving strengths atthe driven sidebands are large, and that the effects of counter-rotating terms are negligible. The deviations arising from aban-doning any of these assumptions may, of course, be calculated.

IX. CONCLUSIONS

We have provided a detailed proposal for configuring athree-mode optomechanical system such that the steady stateincludes a highly pure, highly entangled two-mode squeezedstate. The generation of both mechanical and electromagnetictwo-mode squeezed states has been described. The symmetryof this steady state makes it an attractive platform for theimplementation of continuous-variable teleportation proto-cols. The proposal is efficient in the sense that it requiresonly one driven auxiliary mode to be configured as theengineered reservoir. The problem of unequal single-photonoptomechanical couplings has been overcome by using afour-tone driving scheme, and potential instabilities arisingfrom counter-rotating terms have been accounted for. Asimple experimental signature for the presence of mechanicalentanglement, in terms of the cavity output spectrum, hasbeen provided. The proposal described is implementable withexisting technology.

ACKNOWLEDGMENTS

This work was supported by NSERC, an ECR Grant fromUNSW Canberra, and the DARPA ORCHID program undera grant from the AFOSR. We thank J. Suh, C. U. Lei,I. Petersen, and S. Armstrong for useful discussions.

APPENDIX A: DERIVATION OF HAMILTONIAN

We consider the three-mode optomechanical system de-picted in Figs. 1(a) and 2(a), and start from the Hamiltonian (1)with the four-tone drive (16); the scenario with the two-tonedrive (10) shall be treated as a special case. Moving into arotating frame with respect to H0 = ω1a

†a + ω2b†b + ωcc

†c,recalling Eqs. (18a) and (18b), we obtain

H = �(a†a − b†b) + ga(ae−iω1t + a†e+iω1t )c†c

+ gb(be−iω2t + b†e+iω2t )c†c + Hdrive + Hdiss. (A1)

The effective oscillation frequency � is chosen such thatEqs. (22a) and (22b) are satisfied. The Heisenberg equations,neglecting noise terms, corresponding to Eq. (A1) are

˙a = −i�a − igae+iω1t c†c − γa

2a, (A2a)

˙b = i�b − igbe+iω2t c†c − γb

2b, (A2b)

˙c = −iga(ae−iω1t + a†e+iω1t )c − iE1+e−iω1t

− iE1−e+iω1t − igb(be−iω2t + b†e+iω2t )c

− iE2+e−iω2t − iE2−e+iω2t − κ

2c. (A2c)

Assuming resolved-sideband operation (ωa(b) κ), we takethe ansatz [35]

c(t) = c0(t) + c1+(t)e−iω1t + c1−(t)e+iω1t + c2+(t)e−iω2t + c2−(t)e+iω2t . (A3)

Then, substituting Eq. (A3) into the system (A2a)–(A2c) and separating out the Fourier coefficients of the cavity field we obtainthe system

˙a = −i�a − iga(c†1−c0 + c1+c†0) − γa

2a − igae

+2iδt (c†2−c0 + c2+c†0) − igae

+2iω1t (c†1+c0 + c1−c†0)

− igae+2iωmt (c†2+c0 + c2−c

†0), (A4a)

˙b = i�b − igb(c†2−c0 + c2+c†0) − γb

2b − igbe

−2iδt (c†1−c0 + c1+c†0) − igbe

+2iωmt (c†1+c0 + c1−c†0) − igbe

+2iω2t (c†2+c0 + c2−c†0),

(A4b)

˙c0 = −igaac1− − igaa†c1+ − igbbc2− − igbb

†c2+ − κ

2c0 − iga(ac2−e−2iδt + a†c2+e+2iδt ) − igb(bc1−e+2iδt + b†c1+e−2iδt )

− iga(ac1+e−2iω1t + ac2+e−2iωmt + a†c1−e+2iω1t + a†c2−e+2iωmt ) − igb(bc1+e−2iωmt + bc2+e−2iω2t

+ b†c1−e+2iωmt + b†c2−e+2iω2t ), (A4c)

˙c1− = (−iω1 − κ/2)c1− − igaa†c0 − iE1−, (A4d)

˙c1+ = (iω1 − κ/2)c1− − igaac0 − iE1+, (A4e)

˙c2− = (−iω2 − κ/2)c1− − igbb†c0 − iE2−, (A4f)

˙c2+ = (iω2 − κ/2)c1− − igbbc0 − iE2+. (A4g)

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Note that in writing out Eqs. (A4a)–(A4g) we have re-tained fast-rotating terms only in Eqs. (A4a)–(A4c). SolvingEqs. (A4d)–(A4g) for the steady-state amplitudes of the fieldat the driven sidebands, assuming that the single-photonoptomechanical couplings are relatively small, we obtain theresults of Eq. (17).

Replacing the operator Fourier components at the drivensidebands by their classical steady-state values, we can writean effective (quadratic) Hamiltonian for the system dynamics.If we neglect fast-rotating terms in Eqs. (A4a)–(A4c), thisHamiltonian will be time independent; the time-dependentcontributions to the effective Hamiltonian are given inAppendix D 1.

With four-tone driving, the effective Hamiltonian is (replac-ing c0 → c)

H = �(a†a − b†b) + ga[(c1−a + c1+a†)c† + H.c.]

+ gb[(c2−b + c2+b†)c† + H.c.] + Hdiss. (A5)

Assuming that the drives are matched according to Eq. (19)and that the steady-state amplitudes in the driven sidebandsare real, we get the Hamiltonian (8) with the effectivecouplings of Eq. (20). If there is an imperfection (drive mis-match), then we have the additional Hamiltonian contributionsgiven by Eq. (14) with the coupling imperfections given byEq. (15).

With two-tone driving we have ω1 = ω2 = ωm, and theappropriate effective Hamiltonian is now (A5) with ck±replaced by c±, the driving strengths at the frequenciesωc ± ωm. If the single-photon optomechanical couplings areequal, then the effective Hamiltonian is given by Eq. (8)where the effective couplings are given by Eq. (12). Imper-fections (unequal single-photon optomechanical couplings)lead to additional contributions of the form of Eq. (14),where the effective coupling imperfections are given byEq. (12).

APPENDIX B: TWO-MODE GAUSSIAN STATES

1. Entanglement

The entanglement of a two-mode Gaussian state may bequantified from its symmetrically ordered covariance matrix Vvia the logarithmic negativity [46,47]. Writing the covariancematrix as in Eq. (37), the logarithmic negativity is given by

EN = max {0, − ln 2η} , (B1)

where η = 2−1/2{�(V) − [�(V)2 − 4 det V]1/2}1/2 and�(V) = det Vb + det Va − 2 detVab.

2. Thermal two-mode squeezed state

An alternative method for characterizing the mechanicaltwo-mode state is obtained by noting that the covariancematrix of our steady state takes the form of a thermaltwo-mode squeezed state [48], as defined in Eq. (39). Thisstate is described by three parameters: a two-mode squeezingparameter ξ , and the two initial thermal occupations na

th andnb

th. The covariance matrix for the thermal two-mode squeezedstate, in the block form of Eq. (37), is

Va = caI2, Vb = cbI2, Vab = −cabσz, (B2)

where the coefficients are given by

ca(b) = (n

a(b)th + 1/2

)cosh2 ξ + (

nb(a)th + 1/2

)sinh2 ξ, (B3a)

cab = (na

th + nbth + 1

)sinh ξ cosh ξ. (B3b)

APPENDIX C: HEISENBERG-LANGEVIN EQUATIONS

1. Collective quadratures

In the adiabatic limit and in terms of the collectivequadratures of Eqs. (26a) and (26b), the dynamics of thetwo-mode mechanical system is described by Eq. (44), withthe matrices

A0 =[

A+ A+−A+− A−

], (C1a)

B1 =[

B1+ B1−B1+ B1−

], (C1b)

B2 =√

[I2

022

], (C1c)

including the components

A+ = −(γ /2 + )I2, (C2a)

A− = −(γ /2)I2, (C2b)

B1± =√

γ (1 ± l)/2I2, (C2c)

A+− =[−lγ /2 �

−� −lγ /2

], (C2d)

where l = (γa − γb)/(2γ ) and γ = (γa + γb)/2.

2. Individual quadratures

The dynamics of the linearized, three-mode optomechan-ical system, with Hamiltonian (8), is described by Eq. (46).The system matrix is

A0 =⎡⎣Aa 022 Ca

022 Ab Cb

Ca Cb Ac

⎤⎦, (C3)

where 022 is the 2 × 2 zero matrix, and

Aa =[−γa/2 �

−� −γa/2

], (C4a)

Ab =[−γb/2 −�

� −γb/2

], (C4b)

Ac = −(κ/2)I2, (C4c)

Ca =[

0 G− − G+ − Gms

−G− − G+ + Gmd 0

], (C4d)

Cb =[

0 G− − G+ + Gms

−G− − G+ − Gmd 0

], (C4e)

with the shorthand notation Gms = Gm

− + Gm+ and

Gmd = Gm

− − Gm+. The noise matrix is given by

B0 = Diag(√

γa(na + 1/2)I2,√

γb(nb + 1/2)I2,√

κ/2I2).

(C5)

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M. J. WOOLLEY AND A. A. CLERK PHYSICAL REVIEW A 89, 063805 (2014)

3. Individual mode operators

For the purpose of evaluating the cavity output spectrum,it is more convenient to work in terms of annihilation andcreation operators, as in Eq. (50). The corresponding matricesare

B0 = Diag(√

γaI2,√

γbI2,√

κI2), (C6)

while A0 is given by the block matrix form of Eq. (C3), nowwith

Aa =[−i� − γa/2 0

0 i� − γa/2

], (C7a)

Ab =[i� − γb/2 0

0 −i� − γb/2

], (C7b)

Ca = i

[−G− + Gm− −G+ − Gm

+G+ + Gm

+ G− − Gm−

], (C7c)

Cb = i

[−G− − Gm− −G+ + Gm

+G+ − Gm

+ G− + Gm−

], (C7d)

while Ac is still given by Eq. (C4c).

APPENDIX D: COUNTER-ROTATING CONTRIBUTIONS

1. Hamiltonians

In deriving the time-independent Hamiltonian (8) wediscarded fast-rotating terms; here, we include them. The fulltime-dependent Hamiltonian is

H(t) = H + HCR, (D1)

where H is the time-independent effective Hamiltonian (8) andHCR is the time-dependent (“counter-rotating”) contribution.

a. Four-tone driving

With four driving tones (16), the time-dependent part of the Hamiltonian (D1) is

HCR = ga{a[c2−e−2iδt + c2+e−2iωmt + c1+e−2iω1t ] + a†[c2+e+2iδt + c1−e+2iω1t + c2−e2iωmt ]}c†+ gb{b[c1−e+2iδt + c1+e−2iωmt + c2+e−2iω2t ] + b†[c1+e−2iδt + c2−e+2iω2t + c1−e+2iωmt ]}c† + H.c. (D2)

There are terms at four distinct oscillation frequency magnitudes: the ±2δ terms are associated with the two drives being onthe same side of the cavity resonance frequency, while the terms oscillating at ±2(ωa − �), ± 2ωm, ± 2(ωb + �) are associatedwith two drives on opposing sides of the cavity resonance frequency. In terms of Bogoliubov modes, the Hamiltonian (D2) maybe written as

HCR = Ge−2iδt {β1c†[1/d + (d − 1/d) cosh2 r] − β

†2 c

†(d − 1/d) cosh r sinh r}+Ge+2iδt {β2c

†[1/d − (d − 1/d) sinh2 r] + β†1 c

†(d − 1/d) cosh r sinh r}

−G cosh r sinh r

{β1c

†[e+2iω2t − e−2iω1t + 1

de+i(ω1+ω2)t − de−i(ω1+ω2)t

]

+ β2c†[e+2iω1t − e−2iω2t + de+i(ω1+ω2)t − 1

de−i(ω1+ω2)t

]}

+Gβ†1 c

†[

cosh2 r(e2iω1t + dei(ω1+ω2)t ) − sinh2 r

(1

de−i(ω1+ω2)t + e−2iω2t

)]

+Gβ†2 c

†[

cosh2 r

(e2iω2t + 1

dei(ω1+ω2)t

)− sinh2 r(de−i(ω1+ω2)t + e−2iω1t )

]+ H.c., (D3)

where the asymmetry in the single-photon optomechanicalcoupling rates is parametrized by

d ≡ ga

gb

. (D4)

b. Two-tone driving

With two driving tones, as per Eq. (10), the time-dependentcontribution to the Hamiltonian (D1) is

HCR = G+(a + b)e−2iωmt c†

+G−(a† + b†)e+2iωmt c† + H.c., (D5)

only containing fast-rotating terms oscillating at |2ωm|.

2. Drift matrix

A time-dependent Hamiltonian (D1) leads to a time-dependent drift matrix in the corresponding Heisenberg-Langevin equations [see Eq. (46)]. The drift matrix takes theform given in Eq. (48), and we specify the coefficient matriceshere. In writing out these matrices it is useful to parametrizethe coupling imperfection by ε±, where

c1±c2±

ε± ≡ gb

ga

. (D6)

Having both sets of drives matched according to Eq. (19),and therefore no imperfection in the effective coupling,corresponds to ε± = 1. The drift coefficient matrices, for the

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TWO-MODE SQUEEZED STATES IN CAVITY . . . PHYSICAL REVIEW A 89, 063805 (2014)

general case of four-tone driving, are

A1+ = 1

2

⎡⎣ 022 022 dM+

022 022 N+/d

dM− N−/d 022

⎤⎦ , (D7a)

A2+ = 1

2

⎡⎣022 022 022

022 022 Q+022 Q− 022

⎤⎦ , (D7b)

A3+ = 1

2

⎡⎢⎣

022 022 dQ+022 022 Q+/d

dQ− Q−/d 022

⎤⎥⎦ , (D7c)

A4+ = 1

2

⎡⎢⎣

022 022 Q+022 022 022

Q− 022 022

⎤⎥⎦ , (D7d)

where Ak− = A∗k+, 022 is the 2 × 2 zero matrix, and we have

introduced the notation

M± =[i(∓G−ε+ − G+ε−) G−ε+ − G+ε−−G−ε+ − G+ε− i(∓G−ε+ + G+ε−)

],

(D8a)

N± =[i(±G−ε+ + G+ε−) G−ε+ − G+ε−−G−ε+ − G+ε− i(±G−ε+ − G+ε−)

],

(D8b)

Q± =[i(−G−ε+ ∓ G+ε−) −G−ε+ + G+ε−−G−ε+ − G+ε− i(G−ε+ ∓ G+ε−)

],

(D8c)

with Q± given by Q± with the replacement G± → G±. Thescalar tilde quantities are defined as

G± = G±ε±, (D9a)

ε± = 2(1 + ε±)

(1 + ε+)(1 + ε−). (D9b)

3. Time-dependent drift matrix: Solution

Given the form of the drift matrix (48), we expect thecovariance matrix V, given by the solution of (49), to beoscillatory in the long-time limit. We approximate the solutionvia the covariance matrix ansatz [40]

V(t) = V0 +N∑

k=1

(Vk+e+2iδk t + Vk−e−2iδk t ). (D10)

In general, the solution will contain harmonics of thebare frequencies that appear in Eq. (48), as well as theirsum and difference frequencies. However, the solution thatwe really seek is the dc component of the covariancematrix V0.

The equations of motion (46), with the drift matrix (48),may be written in the frequency domain as

N∑k=1

(Ak+ �X[ω − 2δk] + Ak− �X[ω + 2δk]) + (A0 − iωI6) �X[ω] = −B · �Xin[ω] ≡ �N [ω], (D11)

where we have defined the Fourier transform asF[ω] = ∫ +∞−∞ f (t) e−iωt dt . Now, we form the frequency-dependent state and noise

vectors. These are (2N + 1)-dimensional vectors where N is the number of positive-frequency counter-rotating terms in Eq. (48).The nth elements are �X[ω − 2δ|N+1−n| sgn (N + 1 − n)] and �N [ω − 2δ|N+1−n| sgn (N + 1 − n)], respectively. Subsequently, wecan write the linear system

A[ω] · �X[ω] = �N[ω], (D12)

where

A[ω] =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

A0 − i(ω − 2δN )I6 AN−. . .

.... . . A1−

AN+ . . . A1+ A0 − iωI6 A1− . . . AN−

A1+. . .

.... . .

AN+ A0 − i(ω + 2δN )I6

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

. (D13)

Introducing the noise correlation matrix �i,j [ω,ω′] = 〈Ni[ω]N∗j [ω′] + N∗

j [ω′]Ni[ω]〉/2 leads to

�[ω,ω′] = D0 δ[ω − ω′] +N∑

k=1

(Dk+ δ[ω − ω′ − 2δk] + Dk− δ[ω − ω′ + 2δk]), (D14)

where the matrices are defined by (D0)ii = BBT , (Dn+)N+1−n,N+1 = (Dn+)N+1,N+1+n = BBT for n ∈ {1, . . . ,N}, and Dn− =DT

n+. The indices refer to 6 × 6 blocks in the overall matrix.

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M. J. WOOLLEY AND A. A. CLERK PHYSICAL REVIEW A 89, 063805 (2014)

Solving the linear system (D12) leads to

V[ω,ω′] = A−1[ω]�[ω,ω′](A−1[ω′])†. (D15)

We really want the 6 × 6 central block which we denote V[ω,ω′]. This is given by

V[ω,ω′] = V0δ[ω − ω′] +N∑

k=1

(Vk+[ω]δ[ω − ω′ − 2δk] + Vk−[ω]δ[ω − ω′ + 2δk]), (D16)

where we have the coefficients

V0[ω] = [A−1[ω]D0(A−1[ω])†]6, (D17a)

Vk±[ω] = [A−1[ω]Dk±(A−1[ω ∓ 2δk])†]6. (D17b)

The coefficients in Eq. (D10) follow from

V0 = 1

2π

∫ +∞

−∞V0[ω]dω, (D18a)

Vk± = 1

2π

∫ +∞

−∞Vk±[ω]dω. (D18b)

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